Find all $f$ which satisfies $ f:\mathbb{R}_{\geq0} \rightarrow \mathbb{R}, f(x+y^2) \geq f(x)+y $
Let $P(x,y)$ denote the statement $f(x+y^2) \geq f(x)+y$.
There is no such function. Note that for any $n\in\mathbb{N}$ and $1\leq k\leq n$ we have $$P\left(x+\frac{k}{n},\frac{1}{\sqrt{n}}\right):\ f\left(x+\frac{k}{n}\right)\geq f\left(x+\frac{k-1}{n}\right)+\frac{1}{\sqrt{n}}.$$ Adding up these inequalities we get $f(x+1)\geq f(x)+\sqrt{n}$. For $x$ fixed and $n$ sufficiently large this a contradiction.